Multiply the following complex numbers, marked as blue dots on the graph: $[\cos(\frac{23}{12}\pi) + i \sin(\frac{23}{12}\pi)] \cdot [2(\cos(\frac{5}{3}\pi) + i \sin(\frac{5}{3}\pi))]$ (Your current answer will be plotted in orange.)
Solution: Multiplying complex numbers in polar forms can be done by multiplying the lengths and adding the angles. The first number ( $\cos(\frac{23}{12}\pi) + i \sin(\frac{23}{12}\pi)$ ) has angle $\frac{23}{12}\pi$ and radius $1$ The second number ( $2(\cos(\frac{5}{3}\pi) + i \sin(\frac{5}{3}\pi))$ ) has angle $\frac{5}{3}\pi$ and radius $2$ The radius of the result will be $1 \cdot 2$ , which is $2$ The sum of the angles is $\frac{23}{12}\pi + \frac{5}{3}\pi = \frac{43}{12}\pi$ The angle $\frac{43}{12}\pi$ is more than $2 \pi$ . A complex number goes a full circle if its angle is increased by $2 \pi$ , so it goes back to itself. Because of that, angles of complex numbers are convenient to keep between $0$ and $2 \pi$ $\frac{43}{12}\pi - 2 \pi = \frac{19}{12}\pi$ The radius of the result is $2$ and the angle of the result is $\frac{19}{12}\pi$.